1. Finding an Unpredictable Target in a Workspace with Obstacles LaValle, Lin, Guibas, Latombe, and Motwani, 1997 CS326 Presentation by David Black-Schaffer

2. Overview Searching a complicated environment in such a way that an “evader” can’t “sneak” by. Applies to: adversarial situations, locating items which may move during the search

3. The Strategy Courtesy of Professor Latombe

4. Related Problems Homicidal Chauffeur (no Geometry) –Fast car vs. slow maneuverable human Art Gallery (no Motion) –How many observers needed to cover the whole space? M. Falcone Homicidal ChauffeurArt Gallery

5. Topics Bounds on how many pursuers are needed Information space representation How to find a path

6. Assumptions Target motion is continuous 2D, omnidirectional unlimited distance sensors Evader Pursuer

7. Algorithm Goals A fast, efficient solution strategy Bounds on the number of pursuers needed in terms of the geometry

8. Number of Pursuers Depends on the geometry and topology of the free space Crucial to issues of “completeness” of the algorithm

9. Upper Bounds Simply-connected: n edges, O(lg n) With holes: h holes, n edges: O(lg n + sqrt(h)) Simply-connectedHole

10. Lower Bounds Parson’s Problem: depth k, O(k+1) –Connected graph evasion –Can be converted into corridor with four bends

11. Parson’s Problem

12. Finding a Solution Information Space State Representation Only keep Critical Information Changes

13. Information Space Incomplete knowledge of state –Where is the evader? Work with what we do know and can compute: –Location of the Pursuer –Visibility Region Define our State based on: –Current Free Space location –State of the Free Space Edges at that location (contaminated/clean)

14. Information State 4 possible Information States at this location: –(0,0), (0,1), (1,0), (1,1) By knowing the location in the Free Space and the state of the gap edges we uniquely define the Information State of the system. 1 or 0 (x,y)

15. Key Point Multiple Information Space Points may map to the same Cartesian Point

16. Critical Information Changes Information State only changes when a gap edge appears or disappears. Conservative Cell Partitioning Keep track of just these transitions to simplify without losing completeness. Information State: (x1,y1,0,1)Information State: (x2,y2,0,1)Information State: (x3,y3,0,1)Information State: (x4,y4,0)Information State: (x3,y3,0,0)Information State: (x,y,x, x) Clean Contaminated

17. Partitioning Shoot rays off edges in both directions if possible and from vertices if no collisions in either direction

18. Finding a Path Move between the Free Space centriods of the partitions How to plan a path in Information Space?

19. Information State Graph Connects all possible Information Space States –All edge gap contaminated/clean combinations at all points –A point with 2 edge gaps will have four nodes (00, 01, 10, 11) in this graph –Can grow exponentially Keep track of gap edges splitting or merging –Connections between Information Space States –Number of gaps may change; need to preserve the connectivity –Preserve contamination Search the graph for a solution (Dijksta’s Algorithm) –Initial State will have all contaminated edges (11…) –Goal State will have all clean edges (00…) –Each vertex will only be visited once –Cost function based on Euclidian distance between points

20. Solution Clean Contaminated Visible

21. In More Detail

22. Re-contamination

23. Multiple Pursuers Do one as best you can (greedy algorithm) Add another to cover the missed spaces Less complete, but works pretty well

24. Conclusions Works well in 2D with simple geometry and perfect vision –Fast (a few seconds on a 200MHz RISC machine) –Very effective for cases requiring only 1 robot –Elegant approach

25. However… Requires a simple, 2D geometry –Can simplify more complex geometry –Need to watch out for collisions Information State Graph can be very big Deterministic: not adaptable to partial information Real-world vision is not perfect –Can deal with cone vision

26. 2 Robots Courtesy of Professor Latombe

27. Animated Visibility Courtesy of Professor Latombe